DIFFERENTIALLY SIMPLE RINGS WITH NO INVERTIBLE DERIVATIVES
DIFFERENTIALLY SIMPLE RINGS WITH NO INVERTIBLE DERIVATIVES
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DOI：
10.1093/qmath/32.4.417
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发表时间：
1981-12
影响因子：
0.7
通讯作者：
D. Jordan
中科院分区：
文献类型：
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作者：
D. Jordan
设R是具有非零导子d的交换Noether Q-代数，使得R是d-单的，即R在d下不变的理想只有0和R。众所周知，例如参见[2]，第43页，微分多项式R [x，d]的环是简单的诺特环。KR Goodearl [4]问是否必须有R的元素r使得d（r）是可逆的。本文的目的是证明：如果K是特征为零的域，且n5>> 2，则多项式环R=^[* iX 2，···，* n]允许一个K-导子d，使得R是d-单环且d（R）不含单位元. Goodearl问题背后的动机，除了观察到在所有最著名的d-单环的例子中存在可逆导子之外，是一种感觉，简单的环R [x，d]所产生的一个d-简单的环R没有可逆的衍生物可能具有显着不同的性质，从那些已知的简单环的微分多项式。在这样一个例子R [x，d]和例如Weyl代数At（C）之间可以得出的一个区别是，前者允许一个外导子，而后者则不允许（见[3]，p.148）。任何微分多项式环R [x，d]都有形式微分的导子dldx，且这个导子是内导子当且仅当存在reR使得d（r）= l. K [x1，x2，...，Xn]，我们证明了K [xu* 2»···>* n]^d-简单不是第一个这样的d被构造。J. Archer [1]已经表明，如果K是特征为零的域，则K [xu x2，...，XB]是6-简单的，其中
LET R be a commutative Noetherian Q-algebra with a non-zero derivation d such that R is d-simple, that is the only ideals of R invariant under d are 0 and R. It is well-known, see for example [2], p. 43, that the ring of differential polynomials R [x, d] is a simple Noetherian ring. KR Goodearl [4] has asked whether there has to be an element r of R such that d (r) is invertible. The aim of this note is to show that if K is a field of characteristic zero and n 5» 2 then the polynomial ring R=^[* ii X2,•••,* n] admits a K-derivation d such that R is d-simple and d (R) contains no units.The motivation behind Goodearl's question, apart from the observation that in all the best known examples of d-simple rings there occur invertible derivatives, is a feeling that the simple ring R [x, d] arising from a d-simple ring R with no invertible derivatives might possess significantly different properties from those of the known simple rings of differential polynomials. One distinction which can be drawn between such an example R [x, d] and, say, the Weyl algebra At (C) is that the former admits an outer derivation whereas it is known, see [3], p. 148, that the latter does not. Any ring R [x, d] of differential polynomials admits the derivation dldx of formal differentiation and this derivation is inner if and only if there exists reR such that d (r)= l. The derivation d of K [xlt x2,..., Xn] for which we show that K [xu* 2»•••>* n]^ d-simple is not the first such d to be constructed. J. Archer [1] has shown that, provided K is a field of characteristic zero, K [xu x2,..., XB] is 6-simple where